All Questions
5,882 questions
0
votes
1
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114
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Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 188k
4
votes
4
answers
2k
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I want a smooth orthogonalization process
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 103
2
votes
1
answer
1k
views
The difficulty of generate complex Hadamard matrix
A complex $n\times n$ matrix $A=[a_{ij}]$ is called a Hadamard matrix if $A^{+}A=nI$ and $|a_{ij}|=1$ holds for all $i,j$, where $A^{+}$ denotes the conjugate transposed matrix of $A$, and a vector $...
CommunityBot
- 1
3
votes
2
answers
303
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Asymptotics of A000613
The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
- 148k
2
votes
0
answers
331
views
What is the spectrum of this differential operator?
My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
- 206
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 256
109
votes
15
answers
12k
views
Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...
- 101
3
votes
1
answer
153
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Number of points covered by $2n$ hyperplanes in $\mathbf{F}_p^n$
For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality.
For $s_1,\...
- 148k
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
21
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0
answers
520
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Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
- 1,206
2
votes
1
answer
172
views
Diagonalize almost symmetric tridiagonal matrix
I begin with an $n \times n$ real symmetric tridiagonal matrix. However, I replace the non-zero elements in the first and last rows with zeros, so it is no longer symmetric
$$M = \begin{bmatrix} 0 &...
- 1,538
0
votes
1
answer
127
views
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 20.2k
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
0
votes
0
answers
94
views
Infinite sequence of PSD non-moments in two variables
Define a 2d sequence to be a mapping $a: \mathbb{N}^2 \to \mathbb{R}$ (where $\mathbb{N} = \{0, 1, \dots\}$). Here are two definitions of types of 2d sequences:
We say that a 2d sequence $a$ is a ...
- 161
4
votes
1
answer
190
views
Is the transpose of an infinite Hadamard matrix also Hadamard?
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are almost orthogonal if there is a positive integer $C_0\in \omega$ such that for all $n\in\...
- 9,720
1
vote
0
answers
46
views
Regression models as local sections of a chain complex
Let's say we find some regression equation $\ell$ (best fit / linear / whatever words you need to put here) for a sample $D$, subset of population $P$. This equation/model can be thought of as a ...
- 1,611
15
votes
1
answer
649
views
On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 20.2k
7
votes
1
answer
557
views
Counting points on the intersection of a box and a lattice
Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
CommunityBot
- 1
12
votes
2
answers
1k
views
Stable conjugacy for integer matrices
$\DeclareMathOperator\GL{GL}$Let $F$ be a field, and $E$ an extension field. Then two matrices in $\GL_n(F)$ are conjugate if and only if they are conjugate in $\GL_n(E)$. I'm curious whether the ...
- 63.9k
4
votes
1
answer
103
views
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
- 12.7k
1
vote
1
answer
133
views
Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized
A graceful labeling of a connected simple undirected graph $G=(V,E)$ is a map $f:V\to\lbrace 1,...,|E|+1\rbrace$ such that for all $t\in\lbrace 1,...,|E|\rbrace$ there is a (trivially unique) $\langle ...
- 256
7
votes
1
answer
319
views
The set of strongly positive forms is a closed cone
This question comes from the Complex Analytic and Differential Geometry by Demailly. Let $V$ be a $n$ dimensional complex space. Consider the exterior algebra $\Lambda V^* = \oplus \Lambda^{(p,q)}V^*$....
- 206
1
vote
0
answers
80
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 10.4k
4
votes
1
answer
54
views
Krein-Rutman for integral transforms: proof of convergence to leading eigenvector
Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory.
Consider an integral ...
- 630
2
votes
0
answers
108
views
Largest prime determinant of a binary matrix
Given an integer $n$, I want to prove the existence of an $n\times n$ binary matrix (with 0,1 entries), whose determinant is a prime number. What is a lower bound on the largest determinant that I ...
- 3,549
5
votes
1
answer
303
views
Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 12.9k
2
votes
2
answers
127
views
Optimizing a matrix quadratic form with respect to Loewner order
Fix integers $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times k}$ be such that $P^T P$ has full rank.
Let $\mathcal{X}$ denote the set of unit trace, real $n \times n$ symmetric positive ...
- 20.2k
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
CommunityBot
- 1
1
vote
1
answer
338
views
distance between unitary and anti-unitary matrices
This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might ...
CommunityBot
- 1
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 16.2k
20
votes
1
answer
557
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\...
- 6,353
0
votes
1
answer
397
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
CommunityBot
- 1
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 111
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votes
2
answers
414
views
Recovering eigenvalues of a matrix from its $p$th compound matrix
This question was motivated by Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?. Let $A$ be an $n\times n$ matrix over a field. Suppose
we are given the $p$th ...
CommunityBot
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15
votes
3
answers
1k
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Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 141
0
votes
0
answers
52
views
What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
- 109
1
vote
0
answers
63
views
The rank of a matrix expression
I'm studying discrete-time LTI systems and state estimators for them. Recently, I studied this paper. I am facing a matrix rank calculation problem and having trouble solving it. I will provide more ...
- 4,484
4
votes
1
answer
230
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$\omega\times\omega$-Hadamard matrices
In the following, we define infinite Hadamard matrices.
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\...
- 48.9k
0
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0
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70
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Cyclotomic eigenvalue question for Distance-regular graph
I have read this paper. So, I am just thinking about if the following guess is true:
GUESS: Any Distance-regular graph (DRG) has cyclotomic character value property (which means the eigenvalues of a ...
- 109
1
vote
0
answers
58
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Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
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10
votes
3
answers
2k
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Partial inverse of a matrix - or does it have its own name?
In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...
- 21
2
votes
2
answers
227
views
Is a probabilistic implementation of unitaries invertible?
Let $\{p_j\}_j$ be a set of probabilities, $\sum_j p_j = 1$, let $\{h_j\}_j$ be a set of $n \times n$ Hermitian matrices, and define $ad_h(A) $ be the adjoint.
Define the following linear mapping
$$ E(...
7
votes
3
answers
958
views
Vector of integers such that almost all dot products are positive
Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that
$$
\sum_{j=1}^n a_j\cdot x_j <0
\...
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
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7
votes
4
answers
558
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Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 128k
0
votes
2
answers
252
views
“Smallest” non-zero linear combination of vectors to obtain a non-negative vector
We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form
\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_j \\
\end{bmatrix}
where $x_{i} \geq 0$ for all $i=1,\...
- 651
3
votes
1
answer
202
views
Intermediate lattices $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$
Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer valued matrix (in my case it is the symmetric part of a Cartan matrix). Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank ...
- 4,219
1
vote
1
answer
153
views
How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
- 126
2
votes
0
answers
38
views
Constructing an $n$-simplex at the border of a $n$-ball by orthogonal hyperplanes
I want to construct an $n$-simplex the following way:
Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together.
Place the orthogonal affine $n-1$-...
- 63.9k